The Black-Box Complexity of Nearest Neighbor Search

We define a natural notion of efficiency for approximate nearest-neighbor (ANN) search in general n-point metric spaces, namely the existence of a randomized algorithm which answers (1 + ε)-approximate nearest neighbor queries in polylog(n) time using only polynomial space. We then study which families of metric spaces admit efficient ANN schemes in the black-box model, where only oracle access to the distance function is given, and any query consistent with the triangle inequality may be asked. For ε < 2 5 , we offer a complete answer to this problem. Using the notion of metric dimension defined in [GKL03] (`a la [Ass83]), we show that a metric space X admits an efficient (1+ε)-ANN scheme for any ε < 2 5 if and only if dim(X) = O(log log n). For coarser approximations, clearly the upper bound continues to hold, but there is a threshold at which our lower bound breaks down—this is precisely when points in the “ambient space” may begin to affect the complexity of �...

• Type: Conference
• Year: 2004
• Where: ICALP
• Authors: Robert Krauthgamer, James R. Lee
• Keywords: Efficient Ann Schemes, ICALP 2004, Metric Spaces, N-point Metric Spaces, Theoretical Computer Science